Parity obstructions and additive orthogonal decompositions over the rings Z_2^r

Abstract

We study additive decompositions of matrices over the rings Z₂^rinto sums of orthogonal matrices. Unlike the odd-modulus case, where orthogonal matrices generate the full matrix ring, the even-modulus case exhibits parity obstructions due to the non-invertibility of 2. We establish necessary conditions based on reduction modulo 2, trace parity, and row/column balance. In low dimensions, we recover known obstructions and provide explicit constructions when these conditions are satisfied. We further propose a classification framework based on finitely many Z₂-valued invariants and state related conjectures and open problems.